MIMS EPrints
Not a member yet
2151 research outputs found
Sort by
The uniform companion for large differential fields of characteristic 0
No full textWe show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic , which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields
Symplectic integrators and optimal control
Get PDFWhen selecting a numerical method to integrate an ODE system, it is intuitively clear that preservation of geometric properties is desirable. The particular subclasses of ODE systems we will consider are Lagrangian and Hamiltonian systems. The dynamical equations for these derive from variational principles, and we obtain structure preserving integrators by discretizing the principles rather than the ODEs they generate. We demonstrate some advantages that these symplectic integrators have over methods that are more rudimentary by looking at some examples from optimal control theory.
Our major motivation for considering symplectic integrators is solving an image registration problem, where, using the least effort, we associate a set of landmark points on one image to a corresponding set of points on another. A mathematical formulation of this problem is as a Hamiltonian system; this becomes apparent once we realize that we are computing the motion of particles (the landmark points) under some appropriate potential function. We
investigate the performance of symplectic methods on this, more complex, problem. We show that by formulating the problem as a system of nonlinear equations rather than one of optimal control, the explicit Euler method performs better than the symplectic integrators, especially on a set of data points generated by a real experiment. We give some
evidence that the higher-order methods in Matlab's ODE suite
may be better still, but we do not pursue this line of investigation in any detail
On the American option problem
No full textWe show how the change-of-variable formula with local time on curves derived recently in Peskir (2002) can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in Myneni (1992) and dating back to McKean (1965)
Improving the forward solver for the complete electrode model in EIT using algebraic multigrid
Get PDFImage reconstruction in electrical impedance tomography is an ill-posed nonlinear inverse problem. Linearization techniques are widely used and require the repeated solution of a linear forward problem. To account correctly for the presence of electrodes and contact impedances, the so-called complete electrode model is applied. Implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method. However, preconditioners are essential for efficient convergence. Preconditioners based on incomplete factorization methods are commonly used but their performance depends on user-tuned parameters. To avoid this deficiency, we apply black-box algebraic multigrid, using standard commercial and freely available software. The suggested solution scheme dramatically reduces the time cost of solving the forward problem. Numerical results are presented using an anatomically detailed model of the human head
Magnetohydrodynamic damping of oscillations in low-Prandtl-number convection
Get PDFWe present the results of an experimental investigation of the effect of a magnetic field on the stability of convection in a liquid metal. A rectangular container of gallium is subjected to a horizontal temperature gradient and a uniform magnetic field is applied separately in three directions. The magnetic field suppresses the oscillation most effectively when it is applied in the vertical direction and is least efficient when applied in the direction of the temperature gradient. The critical temperature difference required for the onset of oscillations is found to scale exponentially with the magnitude of the magnetic field for all three orientations. Comparisons are made with available theory and qualitative differences are discussed
Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation
No full textWe consider reversible and \Bbb{Z}_2 -symmetric systems of ordinary differential equations (ODEs) that possess a symmetric homoclinic orbit to a degenerate equilibrium. The equilibrium is supposed to undergo a reversible pitchfork bifurcation, controlled by the system's parameter. It has been shown in Wagenknecht (Nonlinearity 15 2097–119) that a multitude of homoclinic orbits emerges in this bifurcation. In particular, if a coefficient in the normal form of the local bifurcation has the correct sign such that this bifurcation is of eye-type, then globally a reversible homoclinic pitchfork bifurcation can be observed. This means, that similar to the local bifurcation in which two new equilibria emerge, two-homoclinic orbits to these equilibria bifurcate from the primary homoclinic orbit. In this paper, we investigate the emergence of two-homoclinic and two-heteroclinic orbits, that is, orbits making two windings in a neighbourhood of the primary orbit, in this bifurcation. Using a combination of geometrical and analytical techniques we prove the emergence of a family of two-homoclinic orbits to periodic orbits and of a two-heteroclinic cycle between equilibria. The general analysis is illustrated by numerical results for an example system of two second order ODEs
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root
Get PDFFor any matrix automorphism group \G associated with a bilinear
or sesquilinear form, Mackey, Mackey, and Tisseur have recently
shown that the matrix sign decomposition factors of A\in\G also
lie in \G; moreover, the polar factors of lie in \G if the
matrix of the underlying form is unitary. Groups satisfying the
latter condition include the complex orthogonal, real and complex
symplectic, and pseudo-orthogonal groups. This work is concerned
with exploiting the structure of \G when computing the polar and
matrix sign decompositions of matrices in \G. We give sufficient
conditions for a matrix iteration to preserve the group structure
and show that a family of globally convergent rational
Pad\'e-based iterations of Kenney and Laub satisfy these
conditions. The well-known scaled Newton iteration for computing
the unitary polar factor does not preserve group structure, but we
show that the approach of the iterates to the group is precisely
tethered to the approach to unitarity, and that this forces a
different and exploitable structure in the iterates. A similar
relation holds for the Newton iteration for the matrix sign
function. We also prove that the number of iterations needed for
convergence of the structure-preserving methods can be precisely
predicted by running an associated scalar iteration. Numerical
experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping
criteria. The overall conclusion is that the structure-preserving
iterations and the scaled Newton iteration are all of practical
interest, and which iteration is to be preferred is
problem-dependent
Geometric Mechanics and Symmetry: The Peyresq Lectures
No full textThis consists of lecture notes from 6 courses held at 2 summer schools in Peyresq, France in 2000 and 2001. The notes were written up by the lecturers together with some participants
The Conditioning of Linearizations of Matrix Polynomials
No full textThe standard way of solving the polynomial eigenvalue problem of degree
in matrices
is to ``linearize'' to a pencil in matrices
and solve the generalized eigenvalue problem.
For a given polynomial, , infinitely many linearizations exist
and they can have widely varying eigenvalue condition numbers.
We investigate the conditioning of
linearizations from a vector space of pencils
recently identified and studied by
Mackey, Mackey, Mehl, and Mehrmann.
We look for the best conditioned linearization and
compare the conditioning with that of the original polynomial.
Two particular pencils are shown always to be
almost optimal over linearizations in for eigenvalues of
modulus greater than or less
than 1, respectively,
provided that the problem is not too badly scaled
and that the pencils are linearizations.
Moreover, under this scaling assumption,
these pencils are shown to be
about as well conditioned as the original polynomial.
For quadratic eigenvalue problems that are not too heavily damped,
a simple scaling is shown to convert the problem to one that is well scaled.
We also analyze the eigenvalue conditioning
of the widely used first and second companion linearizations.
The conditioning of the first companion linearization relative to that of
is shown to depend on the coefficient matrix norms,
the eigenvalue, and the left \ev s of the linearization and of .
The companion form is found to be potentially much more ill conditioned than
,
but if the 2-norms of the coefficient matrices are all approximately 1
then the companion form and are guaranteed to have similar
condition numbers.
Analogous results hold for the second companion form.
Our results are phrased in terms of both the standard relative condition number
and the condition number of Dedieu and Tisseur for the problem in
homogeneous form,
this latter condition number having the advantage of applying to zero and
infinite eigenvalues